A Metric for the Quantification of Macrosegregation During Alloy Solidification
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OSEGREGATION in metallic alloys is a complex casting defect that is a function of the transport phenomena during processing and is affected by material properties, cooling conditions, and system geometry. One difficulty with exploring the mechanisms underlying the formation of macrosegregation patterns is the quantification of the comparison of large compositional datasets, obtained either experimentally or computationally. Commonly, either full composition fields or specific profiles are reported (e.g., References 1 through 3). These data have been used to visualize macrosegregation and explain the physical phenomena responsible for it, but these visualizations are difficult to compare quantitatively. To aid in such a comparison, a single numerical metric that represents the composition field is frequently calculated. A reliable metric is also useful for quantifying uncertainty propagation, in which the behavior of the metric can be used to understand the probable range of the macrosegregation level as a function of the variation of the process input.[4] One metric that is commonly used for quantifying compositional variation is the macrosegregation number,[5,6]
KYLE FEZI and ALEX PLOTKOWSKI, Graduate Research Assistants, and MATTHEW J. M. KRANE, Associate Professor, are with the School of Materials Engineering, Purdue Center for Metal Casting Research, Purdue University, West Lafayette, IN. Contact e-mail: [email protected] Manuscript submitted August 26, 2015. Article published online March 11, 2016 2940—VOLUME 47A, JUNE 2016
the normalized standard deviation of a Gaussian distribution fitted to the composition field: 12 ZZZ 1 1 2 M¼ ðC C0 Þ dV ; ½1 C0 Vtot V where C0 is the nominal composition, Vtot is the total volume of the domain, and C is the measured or predicted local composition field. The integral in Eq. [1] is approximated as the summation " #12 N 1 1 X 2 ðCi C0 Þ DVi ; ½2 M¼ C0 Vtot i¼1 where DVi is the control volume in which Ci represents the local average composition and N is the number of samples (or control volumes in numerical results). This metric assumes that the composition field is normally distributed about the mean or nominal composition. However, it is quite common for this volume-averaged composition distribution to be asymmetric about this mean, depending on the process parameters and material properties. It is also found that this distribution is skewed to lower compositions for elemental partition coefficients (k) greater than unity and to higher values for k < 1 (Figure 1).[7] As a consequence of the assumptions implicit in Eqs. [1] and [2], the commonly used macrosegregation number is only an accurate depiction of the overall compositional variation for the limiting case of a symmetric, Gaussian distribution. When fitting a Gaussian distribution to a skewed dataset, the longer tail is truncated and the shorter tail is artificially extended. Overall, the fitted distribution tends to overpredict the total amount of METALLURGICAL AND MATERIALS TRANSACTIONS A
macrosegregation and underpredict th
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